Problem: Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{t^2 + 4t}{t^2 + 12t + 32}$
First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 4t}{t^2 + 12t + 32} = \dfrac{(t)(t + 4)}{(t + 8)(t + 4)} $ Notice that the term $(t + 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 4)$ gives: $q = \dfrac{t}{t + 8}$ Since we divided by $(t + 4)$, $t \neq -4$. $q = \dfrac{t}{t + 8}; \space t \neq -4$